Linear Equation Calculator

Two modes, one form. This linear equation calculator solves one-variable equations like ax + b = c right out of the box. Add a second equation, and the same linear equation calculator turns into a 2x2 linear system solver. There is no mode button - the linear equation calculator reads what you typed and picks the mode itself.

• a - x coefficient in ax + b = c. The default example uses 4. Any number works, decimals included.
• b - constant on the left side of ax + b = c. Use -7, minus sign included. The linear equation calculator needs that sign to subtract the right quantity - a linear equation calculator is literal about signs.
• c - the right side of ax + b = c. Use 9. Type those three, and the linear equation calculator takes it from there.
• Click Calculate and the first output row returns x = 4, straight from x = (c - b) / a = (9 - (-7)) / 4. That fraction is the linear equation calculator's one-variable engine.

System mode. Fill the second row - d, e, f for the equation dx + ey = f - and three more outputs appear. First the determinant D = a·e - b·d, then x, then y. Type 1, 1, 12, 2, -1, 3, and the linear equation calculator returns D = -3, x = 5, y = 7. That row is Example 3 below. Leave d, e, f empty, and the linear equation calculator stays in one-variable mode - one form, two tools.

Three outputs come back from the linear equation calculator, each in one plain sentence:

• One-variable solution - the value of x that makes ax + b = c true, the output a linear equation calculator exists to print.
• Determinant D - the go/no-go signal of system mode. Nonzero promises exactly one solution. Zero means infinitely many or none, and Example 4 teaches the reading.
• System solution x, y - the pair satisfying both equations at once, printed x first, y second, the linear equation calculator's vector reading order.

This page is more than a static linear equation calculator - it is also an AI tutor you can ask questions:

• Ask in plain words. "Some number times 4, minus 7, gives 9 - find the number." The AI solves it step by step right in the chat, no typing needed.
• Take a photo of your homework. The AI reads the equation and solves it step by step - one equation or a pair of them.
• Ask about the ideas, not just the number. "Why do I subtract b before dividing?" The AI explains each move, and you can check every number against the linear equation calculator on this page.
• Switch modes mid-question. Start with one equation, then add a second. The linear equation calculator re-runs on the spot, now as a solver for linear equations in two unknowns - no retyping of the first row.
• Keep asking follow-ups. Ask "why?" as often as you need. This linear equation calculator page keeps teaching long after the boxes are empty.

A linear equation asks for the number that balances the two sides. Picture a scale. The left pan holds 4x - 7, and the right pan holds 9. The pans hang level, because whatever x turns out to be, 4x - 7 must weigh exactly 9. Solving the equation means unloading both pans with the same moves until x stands alone. Take 7 from each pan, and 4x = 16 remains - the same subtraction the linear equation calculator performs on paper. Split the left side into 4 equal parts, and one part is 4. That is the whole method. x = 4 is the answer, and the linear equation calculator reaches it by the same two moves in one click.

Formally, a linear equation is one where the unknown appears only to the first power - no x², no square roots. The classic setting is linear equations in one variable, and that is exactly what this page's first mode solves. An 1866 algebra text (Ray's New Higher Algebra) filed such equations under "simple equations." Its chapter opens with equations containing only one unknown quantity. The name has faded; the shape has not:

$ax + b = c$

One algebra move family solves every equation of this shape - the family the linear equation calculator was built for. Subtract b from both sides, so ax = c - b. Then divide by a, and the linear equation calculator's job is done:

$x = \frac{c - b}{a}$

That single fraction is the engine inside the linear equation calculator. For 4, -7, 9 it gives (9 - (-7)) / 4 = 4. The only forbidden case is a = 0. Then no x remains at all, and the equation 0x + b = c is either always true (b = c) or never true (b ≠ c). Example 4 shows what those two look like - and how the linear equation calculator should talk about them.

Here is the system of linear equations definition in one line. It is a set of two or more equations using the same unknowns, solved for values that satisfy every equation at once. Add a second linear equation dx + ey = f and one line of algebra stops being enough. Two unknowns need two conditions. In the linear equation calculator's grid, the first row of fields now reads as ax + by = c. The two rows together form a system of two equations in two unknowns.

The test is the determinant:

$D = ae - bd$

A nonzero D promises exactly one pair (x, y) satisfying both equations - the guarantee the linear equation calculator prints before it solves. Geometrically, that pair is the crossing point of two lines. Example 3 has D = -3, so its two lines cross once. When D = 0, the lines are parallel (no solution) or identical (infinitely many). The linear equation calculator reports that instead of inventing a pair - a straight answer no graphing route gives this cleanly.

When D ≠ 0, Cramer's rule hands the linear equation calculator the pair directly:

$x = \frac{ce - bf}{D}, \qquad y = \frac{af - cd}{D}$

The letters match the six fields one to one. That is exactly why the linear equation calculator uses a, b, c, d, e, f. Type the six coefficients, and the linear system solver mode computes D, x, and y in one click - the linear equation calculator doubling as a 2x2 machine. For systems of three equations and up, the gaussian elimination calculator on this site runs the same elimination logic on a larger grid. And the systems of equations calculator page lays every solving method side by side. If your goal is to solve for linear equations one or two at a time, though, this page's form is the shortest route.

Solve 4x - 7 = 9 - the exact equation the linear equation calculator preloads (a = 4, b = -7, c = 9).

1. Add 7 to both sides: 4x = 9 + 7 = 16. (Subtracting b = -7 and adding 7 are the same move - the linear equation calculator stores b exactly as typed.)
2. Divide both sides by 4: x = 16 / 4 = 4.
3. Check in the original equation: 4(4) - 7 = 16 - 7 = 9, which holds - the check the linear equation calculator runs on every solve.

So x = 4. Type 4, -7, 9 into the form, and the linear equation calculator returns the same 4 from x = (c - b) / a - one click, zero slips. Notice the fraction never cared that b was negative. Subtracting -7 means adding 7, and that sign turn is exactly where hand solutions slip. See the mistakes list below for the rest.

Real homework rarely hands the linear equation calculator a pre-cleaned equation. Solve 2(x - 3) = 8. The linear equation calculator sees the cleaned version; you do the distributing.

1. Distribute the 2: 2x - 6 = 8.
2. Add 6 to both sides: 2x = 14.
3. Divide by 2: x = 7.
4. Check: 2(7 - 3) = 2(4) = 8, which holds - the same verification the linear equation calculator stands behind.

For the form, this equation is 2x + (-6) = 8. Type a = 2, b = -6, c = 8, and the linear equation calculator returns 7.

Now run it backwards. Suppose kx + 5 = 17 has the solution x = 3, and k is the missing number. Substitute the solution: k(3) + 5 = 17, so 3k = 12 and k = 4 - the reverse direction no linear equation calculator is asked for often enough. The linear equation calculator works forward. Feeding it a = 4, b = 5, c = 17 confirms x = 3, which closes the loop. Reverse questions like this ask "which equation produces this answer." It is the same fraction the linear equation calculator uses, read from the other end.

Solve the system x + y = 12 and 2x - y = 3 by hand first. Then let the linear equation calculator confirm it.

1. Line both equations up. Row 1 holds 1, 1, 12; row 2 holds 2, -1, 3.
2. Add the two equations: y cancels, 3x = 15, so x = 5. The linear equation calculator reaches the same x by formula.
3. Substitute back: 5 + y = 12, so y = 7 - back-substitution, the linear equation calculator's own next step after the determinant.
4. Check in the second equation: 2(5) - 7 = 3, which holds. The linear equation calculator treats that check as non-negotiable.

The solution is (x, y) = (5, 7) - the pair the linear equation calculator certifies below. Type 1, 1, 12, 2, -1, 3 into the linear equation calculator. It prints D = (1)(-1) - (1)(2) = -3. Then it returns x = (ce - bf)/D = (12(-1) - 1·3)/(-3) = 5 and y = (af - cd)/D = (1·3 - 12·2)/(-3) = 7. Same pair as the hand solve - computed by formula instead of by elimination. The determinant ran its test first: -3 ≠ 0 promised exactly one crossing before any arithmetic began. The geometry behind it is plain - two straight lines, one crossing point - and a linear equations calculator settles that pair in one click, with algebra alone.

Some linear equations refuse to settle on one number. A linear equation calculator that stayed silent about it would be lying - and this linear equation calculator does not lie. Both degenerate cases come from a = 0:

• Every x is a solution. Take 2(x + 1) = 2x + 2 - an equation in disguise: once you distribute, both sides are the same expression. Expand it: 2x + 2 = 2x + 2. Subtract 2x: 2 = 2 - a true statement no matter what x is. The solution set is all real numbers; any value you plug in checks out.
• No x works. Take x + 1 = x + 3. Subtract x: 1 = 3 - false every time. No solution exists, and the linear equation calculator should not print one.

System mode reports its own degenerate case through the determinant - the linear equation calculator's early-warning light. In x + 2y = 4 with 2x + 4y = 8, D = (1)(4) - (2)(2) = 0. The second equation is just the first one doubled - one identical line, infinitely many solutions. Change the 8 to a 9, and the same D = 0 now means parallel lines and no solution. So when the linear equation calculator shows a zero determinant, read it as information. The answer is every point of one line, or no point at all - never a single pair.

Four slips this linear equation calculator sees every day:

1. Losing the sign of b. In 4x - 7 = 9 the constant b is -7, not 7. The b box needs -7, and the linear equation calculator reads it as minus seven. Type 7 instead, and the linear equation calculator computes (9 - 7) / 4 where you meant (9 + 7) / 4.
2. Dividing half a side. From 4x - 7 = 16, add 7 to both sides first to get 4x = 23, then divide by 4. Dividing only the 4x term by 4 while the -7 stays put is the classic silent error. No linear equation calculator can fix a typo it was never given.
3. Scrambling the second row in system mode. d, e, f means d·x + e·y = f, in that order. Swap e and f, and the determinant test turns into nonsense - the linear equation calculator trusts your column order. Line both equations up as ax + by = c and dx + ey = f before typing.
4. Calling det = 0 a bug. A zero determinant is the linear equation calculator reporting that no unique pair exists - parallel lines or one identical line. Example 4 gives the two-minute reading.

Type carefully, and the linear equation calculator does the rest.